Minimax Kernels for Nonparametric Estimation

SUMMARY The minimax kernels for nonparametric function and its derivative estimates are investigated. Our motivation comes from a study of minimax properties of nonparametric kernel estimates of probability densities and their derivatives. The asymptotic expression of the linear maximum risk is established. The corresponding minimax risk depends on the solutions to a kernel variational problem, called minimax kernels. Further study establishes the general form of minimax kernels with their coeecients determined by a system of nonlinear equations. The detailed properties of these minimax kernels are then discussed. Their explicit expressions are obtained by an algorithm developed in the Appendix. Moreover, the relative eeciencies among the minimax kernels, optimal kernels and Gaussian-based kernels are tabulated.